A note on braids and Parseval's theorem

In 1988 Falk and Randell, based on Arnol'd's 1969 paper on braids, proved that the pure braid groups are residually nilpotent. They also proved that the quotients in the lower central series are free abelian groups. This brief note uses an example to provide evidence for a much stronger statement: that each braid $b$ can be written as an infinite sum $b =\sum_0^\infty b_i$, where each $b_i$ is a linear function of the $i$-th Vassiliev-Kontsevich $Z_i(b)$ invariant of $b$. The example is pure braids on two strands. This leads to solving $e^\tau=q$ for $\tau$ a Laurent series in $q$. We set $\tau = \sum_1^\infty (-1)^{n+1} (q^n - q^{-n})/n$ and use Fourier series and Parseval's theorem to prove $e^\tau=q$. For more than two strands the stronger statement seems to rely on an as yet unstated Plancherel theorem for braid groups, which is likely both to be both and to have deep consequences